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Overwriting output/custom.css
In [2]:
import sys

sys.path.append('./python/')

from mayavi import mlab
from mayavi.tools.sources import vector_field, scalar_field
mlab.options.offscreen = True

import numpy as np

import matplotlib
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import make_axes_locatable
import astropy.units as u
import sunpy.map

import yt
import pysac.yt

%matplotlib inline

font_size = 20
pgf_with_latex = {
    "font.size": font_size,
    "axes.labelsize": font_size,  # LaTeX default is 10pt font.
    "legend.fontsize":font_size,
    "xtick.labelsize": font_size,
    "ytick.labelsize": font_size,
    "savefig.transparent": True
    }

matplotlib.rcParams.update(pgf_with_latex)
WARNING:traits.has_traits:DEPRECATED: traits.has_traits.wrapped_class, 'the 'implements' class advisor has been deprecated. Use the 'provides' class decorator.

Simulations of Magnetohydrodynamic Waves Driven by Photospheric Motions

Stuart J. Mumford

Supervisor: Robertus Erdélyi

Solar Physics & Space Plasma Research Centre (SP2RC), School of Mathematics and Statistics, The University of Sheffield

My Publications


Mumford, S. J. and Erdélyi, R. - Monthly Noticies of the Royal Astronomical Society - March 2015 - Volume 449 Issue 2.
Photospheric Logarithmic Velocity Spirals as MHD Wave Generation Mechanisms

Mumford, S. J., Fedun, V., Erdélyi, R. - The Astrophysical Journal - January 2015 - Volume 799, Issue 1
Generation of Magnetohydrodynamic Waves in Low Solar Atmospheric Flux Tubes by Photospheric Motions

The SunPy Community, Mumford, S. J., Christe, S., Pérez-Suárez, D., et. al - Computational Science and Discovery - January 2015 - Volume 8 Issue 1.
SunPy: Python for Solar Physics

Freij N., Scullion E. M., Nelson C. J., Mumford S. J., Wedemeyer S., and Erdélyi R. - The Astrophysical Journal - July 2014 - Volume 791, Issue 1, p.61
The Detection of Upwardly Propagating Waves Channeling Energy from the Chromosphere to the Low Corona

Gent, F. A., Fedun, V., Mumford, S. J., Erdélyi, R. - Monthly Notices of the Royal Astronomical Society - October 2013 - Volume 435, Issue 1, p.689-697
Magnetohydrostatic equilibrium - I. Three-dimensional open magnetic flux tube in the stratified solar atmosphere

Nelson, C. J., Doyle, J. G., Erdélyi, R., Huang, Z., Madjarska, M. S., Mathioudakis, M., Mumford, S. J., Reardon, K - Solar Physics - April 2013 - Volume 283, Issue 2, p.307-323.
Statistical Analysis of Small Ellerman Bomb Events

Problem - Coronal Heating


The solar atmosphere is too hot when compared to known energy input. What are the unknown heating mechanisms?

  • Magnetic Reconnection
  • Magnetohydrodynamic (MHD) Waves

Heating by MHD Waves

  • Waves generated in high-energy low atmosphere.
  • Wave propagate through the atmosphere, along magnetic field lines.
  • Waves deposit energy higher in the atmosphere.

The mechanism by which the wave energy is converted into atmospheric heating in the high atmosphere is unknown, but the properties of the wave behavior is dependant on the wave mode.

How are these waves generated and what are their properties?

Numerical Simulations of the low-atmosphere

The Code

The code used is the Sheffield Advanced Code (SAC) (Shelyag, Fedun, and Erdélyi 2008).

SAC simulates pertabations on a static background, using a CD4 solver with hyper-diffusion and hyper-viscosity terms to stabalise the solution.

This makes it well suited to solving wave pertubations on top of a highly stratified background such as the solar atmosphere.

The Model

To simulate wave excitation in the photosphere a numerical model of the solar atmosphere is needed.

Hydrostatic background from the VAL 3C model (Vernazza, Avrett, and Loeser 1981):

In [3]:
import pysac.mhs_atmosphere as atm

#Read in the VAL3C model
empirical_data = atm.hs_atmosphere.read_VAL3c_MTW(MTW_file=False)


# Create a Z array at the interpolated resolution and interpolate.
ZZ = u.Quantity(np.linspace(empirical_data['Z'][0], empirical_data['Z'][-1], 128), unit=empirical_data['Z'].unit)
table = atm.hs_atmosphere.interpolate_atmosphere(empirical_data, ZZ, s=0)


# Create a figure and make space for the axes.
fig, ax = plt.subplots(gridspec_kw={'right':0.77, 'left':0.16, 'bottom':0.13}, figsize=(13,8))

# Shortcut all the Mm conversion.
Z = empirical_data['Z'].to(u.Mm)

lrho, = ax.plot(Z, empirical_data['rho'].quantity.si, 'x', color='blue')
lrho_i, = ax.plot(ZZ.to(u.Mm), table['rho'].quantity.si, color='blue')

ax2 = ax.twinx()
lp, = ax2.plot(Z, empirical_data['p'].to(u.Pa), 'x', color='green')
lp_i, = ax2.plot(ZZ.to(u.Mm), table['p'].to(u.Pa), color='green')


ax3 = ax.twinx()
ax3.spines["right"].set_position(("axes", 1.2))
lt, = ax3.plot(Z, empirical_data['T'].to(u.K), 'x', color='red')
lt_i, = ax3.plot(ZZ.to(u.Mm), table['T'].to(u.K), color='red')


# Set primary axes properties and labels
ax.semilogy()
ax.set_ylabel(r"Density [{}]".format(lrho._yorig.unit._repr_latex_()))
ax.set_xlabel(r"Height [{}]".format(lrho._xorig.unit._repr_latex_()))
ax.set_xlim(Z[0].value-0.1, Z[-1].value+0.1)


# Pressure Axis
ax2.semilogy()
ax2.set_ylabel(r"Pressure [{}]".format(lp._yorig.unit._repr_latex_()))


# Temp axis
ax3.set_ylabel(r"Temperature [{}]".format(lt._yorig.unit._repr_latex_()))

ax.set_xlim([-0.02,1.62])
ax3.set_ylim([3500,7500])
# Set the colours for the ticks and the labels.
ax.tick_params(axis='y', colors=lrho.get_color())
ax2.tick_params(axis='y', colors=lp.get_color())
ax3.tick_params(axis='y', colors=lt.get_color())

ax.yaxis.label.set_color(lrho.get_color())
ax2.yaxis.label.set_color(lp.get_color())
ax3.yaxis.label.set_color(lt.get_color())
fig
Out[3]:
In [4]:
from pysac.mhs_atmosphere.parameters.model_pars import mfe_setup as model_pars
import pysac.mhs_atmosphere as atm

# Cheeky Reset to Photosphere
model_pars['xyz'][4] = 0*u.Mm
#==============================================================================
# Build the MFE flux tube model using pysac
#==============================================================================
# model setup
scales, physical_constants = atm.units_const.get_parameters()
option_pars = atm.options.set_options(model_pars, False, l_gdf=True)
coords = atm.model_pars.get_coords(model_pars['Nxyz'], u.Quantity(model_pars['xyz']))

#interpolate the hs 1D profiles from empirical data source[s]
empirical_data = atm.hs_atmosphere.read_VAL3c_MTW(mu=physical_constants['mu'])
table = atm.hs_atmosphere.interpolate_atmosphere(empirical_data, coords['Zext'])
table['rho'] += table['rho'].min()*3.6

# calculate 1d hydrostatic balance from empirical density profile
# the hs pressure balance is enhanced by pressure equivalent to the
# residual mean coronal magnetic pressure contribution once the magnetic
# field has been applied
magp_meanz = np.ones(len(coords['Z'])) * u.one
magp_meanz *= model_pars['pBplus']**2/(2*physical_constants['mu0'])

# Make the vertical profile 3D
pressure_z, rho_z, Rgas_z = atm.hs_atmosphere.vertical_profile(coords['Z'], table, magp_meanz,
                                                               physical_constants, coords['dz'])

# Generate 3D coordinate arrays
x, y, z = u.Quantity(np.mgrid[coords['xmin']:coords['xmax']:1j*model_pars['Nxyz'][0],
                              coords['ymin']:coords['ymax']:1j*model_pars['Nxyz'][1],
                              coords['zmin']:coords['zmax']:1j*model_pars['Nxyz'][2]], unit=coords['xmin'].unit)

# Get default MFE flux tube parameters out of pysac
xi, yi, Si = atm.flux_tubes.get_flux_tubes(model_pars, coords, option_pars)

# Generate the 3D magnetic field
allmag = atm.flux_tubes.construct_magnetic_field(x, y, z, xi[0], yi[0], Si[0], model_pars, option_pars,
                                                 physical_constants, scales)
pressure_m, rho_m, Bx, By ,Bz, Btensx, Btensy = allmag

# local proc 3D mhs arrays
pressure, rho = atm.mhs_3D.mhs_3D_profile(z, pressure_z, rho_z, pressure_m, rho_m)
magp = (Bx**2 + By**2 + Bz**2) / (2.*physical_constants['mu0'])
energy = atm.mhs_3D.get_internal_energy(pressure, magp, physical_constants)

#### YT STUFF ####

magnetic_field_x = lambda field, data: data['mag_field_x']
yt.add_field(("gas","magnetic_field_x"), function=magnetic_field_x, units=yt.units.T.units)
magnetic_field_y = lambda field, data: data['mag_field_y']
yt.add_field(("gas","magnetic_field_y"), function=magnetic_field_y, units=yt.units.T.units)
magnetic_field_z = lambda field, data: data['mag_field_z']
yt.add_field(("gas","magnetic_field_z"), function=magnetic_field_z, units=yt.units.T.units)

# Add derived Fields
def magnetic_field_strength(field, data):
    return np.sqrt(data["mag_field_x"]**2 + data["mag_field_y"]**2 + data["mag_field_z"]**2)
yt.add_field(("gas","magnetic_field_strength"), function=magnetic_field_strength, units=yt.units.T.units)

#def alfven_speed(field, data):
#    return np.sqrt(2.*data['magnetic_pressure']/data['density'])
#yt.add_field(("gas","alfven_speed"), function=alfven_speed, units=(yt.units.m/yt.units.s).units)

bbox = u.Quantity([u.Quantity([coords['xmin'], coords['xmax']]),
                   u.Quantity([coords['ymin'], coords['ymax']]),
                   u.Quantity([coords['zmin'], coords['zmax']])]).to(u.m).value

# Now build a yt DataSet with the generated data:
data = {'mag_field_x':yt.YTQuantity.from_astropy(Bx.decompose()),
        'mag_field_y':yt.YTQuantity.from_astropy(By.decompose()),
        'mag_field_z':yt.YTQuantity.from_astropy(Bz.decompose()),
        'pressure': yt.YTQuantity.from_astropy(pressure.decompose()),
        'magnetic_pressure': yt.YTQuantity.from_astropy(magp.decompose()),
        'density': yt.YTQuantity.from_astropy(rho.decompose())}

ds = yt.load_uniform_grid(data, x.shape, length_unit='m', magnetic_unit='T',
                          mass_unit='kg', periodicity=[False]*3, bbox=bbox)
/home/stuart/GitHub/SWAT/pysac/pysac/mhs_atmosphere/mhs_model/flux_tubes.py:312: Warning: pbbal.max() = -0.111891563752 kg / (m s2)
  warnings.warn("pbbal.max() = {}".format(pbbal.max().decompose()), Warning)
yt : [INFO     ] 2015-12-01 23:23:41,518 Parameters: current_time              = 0.0
yt : [INFO     ] 2015-12-01 23:23:41,518 Parameters: domain_dimensions         = [129 129 128]
yt : [INFO     ] 2015-12-01 23:23:41,519 Parameters: domain_left_edge          = [-1000000. -1000000.        0.]
yt : [INFO     ] 2015-12-01 23:23:41,520 Parameters: domain_right_edge         = [ 1000000.   1000000.   1587786.3]
yt : [INFO     ] 2015-12-01 23:23:41,521 Parameters: cosmological_simulation   = 0.0

The Magnetic Flux Tube

The magnetic field model follows (Gent et al. 2013) and is constructed as a self-similar, cylindrically symmetric, expanding field.

In [5]:
slc = yt.SlicePlot(ds, 'x', 'density', origin='lower-center-domain', axes_unit='Mm')
slc.set_figure_size(10)
slc.set_cmap('all', 'viridis')
slc.set_font_size(20)
slc.set_zlim('all', 0, 2.5e-4)


seed_points = np.zeros([11,2]) + 1.52
seed_points[:,0] = np.linspace(-0.99, 0.95, seed_points.shape[0], endpoint=True)
slc.annotate_streamlines('mag_field_y', 'mag_field_z', field_color='magnetic_field_strength',
plot_args={'start_points':seed_points, 'density':15, 'cmap':'Blues', 'linewidth':2,
'norm': matplotlib.colors.LogNorm(*ds.all_data().quantities.extrema("magnetic_field_strength"))})
#force render
slc.save('/tmp/test.png')
#use the raw Figure for transparent bg
slc.plots['density'].figure
yt : [INFO     ] 2015-12-01 23:23:41,602 Loading field plugins.
yt : [INFO     ] 2015-12-01 23:23:41,603 Loaded angular_momentum (8 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,603 Loaded astro (15 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,604 Loaded cosmology (22 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,605 Loaded fluid (63 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,607 Loaded fluid_vector (95 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,608 Loaded geometric (111 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,609 Loaded local (115 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,610 Loaded magnetic_field (121 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,611 Loaded my_plugins (121 new fields)
yt : [INFO     ] 2015-12-01 23:23:41,612 Loaded species (123 new fields)
yt : [INFO     ] 2015-12-01 23:23:42,001 xlim = -1000000.000000 1000000.000000
yt : [INFO     ] 2015-12-01 23:23:42,001 ylim = 0.000000 1587786.300000
yt : [INFO     ] 2015-12-01 23:23:42,004 xlim = -1000000.000000 1000000.000000
yt : [INFO     ] 2015-12-01 23:23:42,005 ylim = 0.000000 1587786.300000
yt : [INFO     ] 2015-12-01 23:23:42,015 Making a fixed resolution buffer of (('gas', 'density')) 800 by 800
yt : [WARNING  ] 2015-12-01 23:23:42,068 Plot image for field ('gas', 'density') has both positive and negative values. Min = -0.000032, Max = 0.000267.
yt : [WARNING  ] 2015-12-01 23:23:42,069 Switching to symlog colorbar scaling unless linear scaling is specified later
yt : [INFO     ] 2015-12-01 23:23:43,692 Saving plot /tmp/test.png
Out[5]:

Exciting Waves in the Photosphere

In [6]:
x_range, y_range = [-300, 300]*u.arcsec, [-250, 250]*u.arcsec

plt.ioff()
fig = plt.figure(dpi=50, figsize=(11,8))
mm = sunpy.map.Map('/home/stuart/VivaData/gband_data/Gband_cospatial_cotemporal_00000.fits').submap(x_range, y_range)
mm = mm.submap([-440,440]*u.arcsec, [-440,440]*u.arcsec)
mm.plot_settings['cmap'] = 'gray'
mm.plot_settings['title'] = ''
im = mm.plot()
fig.savefig('./images/gband-plot.png', transparent=True)

The dynamic photosphere with embedded magnetic field provides many potential ways of driving MHD waves.

  • 'Buffeting'
  • Convective motions (vertically).
  • Spiralling in downdrafts.

Driving Waves in the Simulation Domain

$$ V(x,y,z) = F(x,y,z) \ e^{-\left(\frac{z^2}{\Delta z^2} + \frac{x^2}{\Delta x^2} + \frac{y^2}{\Delta y^2}\right)} \sin \left(2\pi \frac{t}{P}\right) $$

Identifying Waves from Broadband Drivers

Photospheric drivers excite multiple wave modes simulatenously.

How to quantify the relative strengths of the different modes from different drivers.


Assume uniform media:

  • Decompose pertubations into Fast, Slow and Alfvén modes.
  • Compare the percentage wave energy in each mode.
In [7]:
#Define tvtk notebook viewer
from IPython.core.display import Image
import subprocess
def mlab_view(scene, azimuth=153, elevation=62, distance=400, focalpoint=np.array([  25.,   63.,  60.]), aa=16):
    scene.anti_aliasing_frames = aa
    mlab.view(azimuth=azimuth, elevation=elevation, distance=distance, focalpoint=focalpoint)
    scene.save('offscreen.png', size=(750, 750))
    subprocess.call(["convert", "offscreen.png", "-transparent", "white", "offscreen.png"])
    return Image(filename='offscreen.png') 

Relationship between Mode and Velocity Pertubation

Assuming parallel propagation (along the magnetic field) $k_\parallel >> k_\perp$ it can be shown that the slow mode and fast mode perturb different components of velocity:

Fast mode: $$ \frac{|v_\parallel|}{|v_\perp|+|v_\parallel|} = \frac{\frac{k_\parallel}{k_\perp}|v_\perp|}{|v_\perp|+\frac{k_\parallel}{k_\perp}|v_\perp|}\\ = \frac{\frac{k_\parallel}{k_\perp}}{1+\frac{k_\parallel}{k_\perp}}\\ = \frac{1}{\frac{k_\perp}{k_\parallel}+1}\\ \ \\ |v_\parallel| >> |v_\perp| $$
Slow mode: $$ \frac{|v_\parallel|}{|v_\perp|+|v_\parallel|} = \frac{\frac{k_\perp}{k_\parallel}|v_\perp|}{|v_\perp|+\frac{k_\perp}{k_\parallel}|v_\perp|}\\ = \frac{\frac{k_\perp}{k_\parallel}}{1+\frac{k_\perp}{k_\parallel}}\\ = \frac{1}{\frac{k_\parallel}{k_\perp} + 1}\\ \ \\ |v_\parallel| << |v_\perp| $$

Decomposing Velocity Pertubation in SAC

In [8]:
ds = pysac.yt.SACGDFDataset('/home/stuart/VivaData/Slog_p240-0_A10_B005_00001.gdf')
yt : [WARNING  ] 2015-12-01 23:23:45,487 'field_units' was overridden by 'dataset_units/density_bg'
yt : [WARNING  ] 2015-12-01 23:23:45,488 'field_units' was overridden by 'dataset_units/density_pert'
yt : [WARNING  ] 2015-12-01 23:23:45,492 'field_units' was overridden by 'dataset_units/internal_energy_bg'
yt : [WARNING  ] 2015-12-01 23:23:45,494 'field_units' was overridden by 'dataset_units/internal_energy_pert'
yt : [WARNING  ] 2015-12-01 23:23:45,497 'field_units' was overridden by 'dataset_units/mag_field_x_bg'
yt : [WARNING  ] 2015-12-01 23:23:45,499 'field_units' was overridden by 'dataset_units/mag_field_x_pert'
yt : [WARNING  ] 2015-12-01 23:23:45,500 'field_units' was overridden by 'dataset_units/mag_field_y_bg'
yt : [WARNING  ] 2015-12-01 23:23:45,502 'field_units' was overridden by 'dataset_units/mag_field_y_pert'
yt : [WARNING  ] 2015-12-01 23:23:45,504 'field_units' was overridden by 'dataset_units/mag_field_z_bg'
yt : [WARNING  ] 2015-12-01 23:23:45,506 'field_units' was overridden by 'dataset_units/mag_field_z_pert'
yt : [WARNING  ] 2015-12-01 23:23:45,513 'field_units' was overridden by 'dataset_units/velocity_x'
yt : [WARNING  ] 2015-12-01 23:23:45,515 'field_units' was overridden by 'dataset_units/velocity_y'
yt : [WARNING  ] 2015-12-01 23:23:45,518 'field_units' was overridden by 'dataset_units/velocity_z'
yt : [INFO     ] 2015-12-01 23:23:45,556 Parameters: current_time              = 1.00339395144
yt : [INFO     ] 2015-12-01 23:23:45,557 Parameters: domain_dimensions         = [128 128 128]
yt : [INFO     ] 2015-12-01 23:23:45,559 Parameters: domain_left_edge          = [  781250.    781250.   3664122.1]
yt : [INFO     ] 2015-12-01 23:23:45,561 Parameters: domain_right_edge         = [  1.99218750e+08   1.99218750e+08   1.58778630e+08]
yt : [INFO     ] 2015-12-01 23:23:45,563 Parameters: cosmological_simulation   = 0.0
In [9]:
from tvtk.api import tvtk

#pysac imports
import pysac.yt
import pysac.analysis.tube3D.tvtk_tube_functions as ttf
import pysac.plot.mayavi_plotting_functions as mpf
from pysac.plot.mayavi_seed_streamlines import SeedStreamline
from pysac.plot.divergingcolourmaps import get_mayavi_colourmap
from pysac.analysis.tube3D.process_utils import get_yt_mlab

### Load in and Config ###

# loaded above
ds = pysac.yt.SACGDFDataset('/home/stuart/VivaData/Slog_p240-0_A10_B005_00001.gdf')
tube_r = 60

#if running this creates a persistant window just get it out of the way!
mlab.options.offscreen = True
fig = mlab.figure(bgcolor=(1, 1, 1))

cg = ds.index.grids[0]

#Slices
cube_slice = np.s_[:,:,:-5]
x_slice = np.s_[:,:,:,:-5]

#Define the size of the domain
xmax, ymax, zmax = np.array(cg['density'].to_ndarray()[cube_slice].shape) - 1
domain = {'xmax':xmax, 'ymax':ymax, 'zmax':zmax}

bfield, vfield = get_yt_mlab(ds, cube_slice, flux=False)

#Create a scalar field of the magntiude of the vector field
bmag = mlab.pipeline.extract_vector_norm(bfield, name="Field line Normals")
yt : [INFO     ] 2015-12-01 23:23:45,831 Loading field plugins.
yt : [INFO     ] 2015-12-01 23:23:45,833 Loaded angular_momentum (8 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,835 Loaded astro (15 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,837 Loaded cosmology (22 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,839 Loaded fluid (63 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,841 Loaded fluid_vector (95 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,845 Loaded geometric (111 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,846 Loaded local (115 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,847 Loaded magnetic_field (120 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,848 Loaded my_plugins (120 new fields)
yt : [INFO     ] 2015-12-01 23:23:45,850 Loaded species (122 new fields)
In [10]:
xc = domain['xmax']/2
yc = domain['ymax']/2
ti = 0
n = 100

surf_seeds = []
for theta in np.linspace(0, 2 * np.pi, n, endpoint=False):
    surf_seeds.append([tube_r * np.cos(theta + 0.5 * ti) + xc,
    tube_r * np.sin(theta + 0.5 * ti) + yc, domain['zmax']])

seeds = np.array(surf_seeds)
#Add axes:
axes, outline = mpf.add_axes(np.array(zip(ds.domain_left_edge,ds.domain_right_edge)).flatten()/1e8)

#Add seed points to plot:
seed_points = mlab.points3d(seeds[:,0], seeds[:,1], seeds[:,2],
color=(0.231, 0.298, 0.752), scale_mode='none',
scale_factor=1.5)

mlab_view(fig.scene)
/opt/miniconda/envs/thesis/lib/python2.7/site-packages/mayavi/tools/camera.py:288: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison
  if focalpoint is not None and not focalpoint == 'auto':
Out[10]:
In [11]:
field_lines = SeedStreamline(seed_points = np.array(seeds))
bmag.add_child(field_lines)
field_lines.actor.mapper.scalar_visibility = False
field_lines.actor.property.color = (0,0,0)
field_lines.actor.property.line_width = 1.5

mlab_view(fig.scene)
Out[11]:
In [12]:
pd_seeds = ttf.make_circle_seeds(100, 60, **domain)
fieldlines, surface = ttf.create_flux_surface(bfield.outputs[0], pd_seeds)
surface.output.lines = None
flux_surface = mlab.pipeline.surface(surface.output)
flux_surface.actor.mapper.scalar_visibility = False
flux_surface.actor.property.color = (0.8,0.8,0.8)
#flux_surface.actor.property.line_width = 0

mlab_view(fig.scene)
Out[12]:
In [13]:
axes.visible = False
outline.visible = False
flux_surface.actor.property.edge_visibility = True
mlab_view(fig.scene, azimuth = 90, elevation = 75, distance=80, focalpoint=[63, 120, 110], aa=20)
Out[13]:
In [14]:
poly_norms = ttf.make_poly_norms(surface.output)
normvec = mlab.pipeline.glyph(poly_norms.output)
normvec.glyph.glyph_source.glyph_source = normvec.glyph.glyph_source.glyph_dict['arrow_source']
normvec.glyph.glyph.scale_mode = 'data_scaling_off'
normvec.glyph.glyph.color_mode = 'color_by_scale'
normvec.glyph.glyph.scale_factor = 5
normvec.glyph.glyph_source.glyph_position = 'tail'

mlab_view(fig.scene, azimuth=85, elevation=80, distance=50, focalpoint=[63, 120, 110], aa=20)
Out[14]:

MHD Waves excited by Different Photospheric Drivers

Chapter 4




Mumford, S. J., Fedun, V., Erdélyi, R. - The Astrophysical Journal - January 2015 - Volume 799, Issue 1
Generation of Magnetohydrodynamic Waves in Low Solar Atmospheric Flux Tubes by Photospheric Motions

In [15]:
from streamlines import Streamlines
#Use Equation 1 to calculate the vector field in a 2D plane to plot it.
time = np.linspace(0,60,480)
dt = time[1:] - time [:-1]
period = 240.

x = np.linspace(7812.5,1992187.5,128)
y = np.linspace(7812.5,1992187.5,128)

x_max = x.max()
y_max = y.max()

xc = 1.0e6
yc = 1.0e6

xn = x - xc
yn = y - yc

delta_x=0.1e6
delta_y=0.1e6

xx, yy = np.meshgrid(xn,yn)
exp_y = np.exp(-(yn**2.0/delta_y**2.0))
exp_x = np.exp(-(xn**2.0/delta_x**2.0))

exp_x2, exp_y2= np.meshgrid(exp_x,exp_y)
exp_xyz = exp_x2 * exp_y2


#==============================================================================
# Define Driver Equations and Parameters
#==============================================================================
#A is the amplitude, B is the spiral expansion factor
A = 10

#Tdamp defines the damping of the driver with time, Tdep is the ocillator
tdamp = lambda time1: 1.0 #*np.exp(-(time1/(period)))
tdep = lambda time1: np.sin((time1*2.0*np.pi)/period) * tdamp(time1)

#Define a peak index to use for scaling in the inital frame
max_ind = np.argmax(tdep(time) > 0.9998)

def log():
    B = 0.05
    phi = np.arctan2(1,B)
    theta = np.arctan2(yy,xx)

    uy = np.sin(theta + phi)
    ux =  np.cos(theta + phi)

    vx = lambda time1: (ux / np.sqrt(ux**2 + uy**2)) * exp_xyz * tdep(time1) * A
    vy = lambda time1: (uy / np.sqrt(ux**2 + uy**2)) * exp_xyz * tdep(time1) * A

    vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)

    return vx, vy, vv

def arch():
    B = 0.005
    r = np.sqrt(xx**2 + yy**2)

    vx = lambda time1: ( (B*1e6 * xx) / (xx**2 + yy**2) + yy/r ) * exp_xyz * tdep(time1) * A
    vy = lambda time1: ( (B*1e6 * yy) / (xx**2 + yy**2) - xx/r ) * exp_xyz * tdep(time1) * A

    vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)

    return vx, vy, vv

def uniform():
    #Uniform
    vx = lambda time1: A * (yy / np.sqrt(xx**2 + yy**2)) * exp_xyz * tdep(time1)
    vy = lambda time1: A * (-xx / np.sqrt(xx**2 + yy**2)) * exp_xyz * tdep(time1)
    vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)
    
    return vx, vy, vv

drivers = [log, arch, uniform]

#fig, axs = plt.subplots(1, 3, figsize=(18,9))

for driver_func in drivers:
    fig, ax = plt.subplots(figsize=(7,6), dpi=300)
    #============================================================================
    # Do the Plotting
    #============================================================================
    vx, vy, vv = driver_func()
    # Calculate Streamline
    slines = Streamlines(x,y,vx(time[max_ind]),vy(time[max_ind]),maxLen=7000,
                         x0=xc, y0=yc, direction='forwards')

    im = ax.imshow(vv, cmap='Blues', extent=[7812.5,x_max,7812.5,y_max])
    im.set_norm(matplotlib.colors.Normalize(vmin=0,vmax=A))
    #ax.hold()
    
    if driver_func != uniform:
        Sline, = ax.plot(slines.streamlines[0][0],slines.streamlines[0][1],color='red',linewidth=2, zorder=40)
    else:
        Sline = matplotlib.patches.Circle([1e6, 1e6], radius=.15e6, fill=False, color='red', linewidth=2, zorder=40)
        ax.add_artist(Sline)

    #Add colourbar
    divider = make_axes_locatable(ax)
    cax = divider.append_axes("right", size="5%", pad=0.2)
    cbar = plt.colorbar(im,cax)
    cbar.set_label(r"$|V|$ [ms$^{-1}$]")
    scalar = matplotlib.ticker.ScalarFormatter(useMathText=False,useOffset=False)
    scalar.set_powerlimits((-3,3))
    cbar.formatter = scalar
    cbar.ax.yaxis.get_offset_text().set_visible(True)
    cbar.update_ticks()
    #cbar.solids.set_rasterized(True)
    cbar.solids.set_edgecolor("face")

    #Add quiver plot overlay
    qu = ax.quiver(x, y, vx(time[max_ind]), vy(time[max_ind]), scale=25*A, color='k', zorder=20, linewidth=1)
    ax.axis([8.0e5,12.0e5,8.0e5,12.0e5])

    ax.xaxis.set_major_formatter(scalar)
    ax.yaxis.set_major_formatter(scalar)
    ax.xaxis.set_major_locator(matplotlib.ticker.MaxNLocator(5))
    ax.yaxis.set_major_locator(matplotlib.ticker.MaxNLocator(5))
    ax.xaxis.get_offset_text().set_visible(False)
    ax.yaxis.get_offset_text().set_visible(False)
    ax.set_xlabel("X [Mm]")
    ax.set_ylabel("Y [Mm]")

    fig.savefig('images/driver_{}.png'.format(driver_func.__name__), transparent=True)

Driving Waves in the Simulation Domain


$$ V_{(x,y,z)} = F_{(x,y,z)} \ e^{-\left(\frac{z^2}{\Delta z^2} + \frac{x^2}{\Delta x^2} + \frac{y^2}{\Delta y^2}\right)} \sin \left(2\pi \frac{t}{P}\right) $$

Uniform Driver

$$ F(x) = A \frac{y}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{x}{\sqrt{x^2 + y^2}}, $$

Archmedian Spiral


$$ F(x) = A \frac{B_Ax}{x^2 + y^2} \frac{y}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{B_Ay}{x^2 + y^2} \frac{x}{\sqrt{x^2 + y^2}}. $$

Logarithmic Spiral


$$ F(x) = A \frac{\cos(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{\sin(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ $$ where, $\theta = \tan^{-1}\left(\frac{y}{x}\right),\ \phi = \tan^{-1}\left(\frac{1}{B_L}\right)$

Analysis and Results

Maybe the YouTube vido here, or some other vid if I can find / make one.

Wave Flux

Calulate wave energy flux from (Leroy 1985). $$ \vec{F}_{\textbf{wave}} \equiv \widetilde{p}_k \vec{v} + \frac{1}{\mu_0} \left(\vec{B}_b \cdot \vec{\widetilde{B}}\right) \vec{v} - \frac{1}{\mu_0}\left(\vec{v} \cdot \vec{\widetilde{B}} \right) \vec{B}_b $$

Time-Distance Diagrams

In [16]:
import td_plotting_helpers as ph
import time_distance_plots as tdp
from matplotlib.image import NonUniformImage
import texfigure
ch4 = texfigure.Manager(None, number=4, base_path='/home/stuart/GitHub/Thesis/thesis/Chapter4/')

figsize = (17,8)

pvel_labels = {'par_label':r'$V_\parallel[$ ms$^{-1}$]', 
                'perp_label':r'$V_\perp$ [ms$^{-1}$]',
                'phi_label':r'$V_\phi$ [ms$^{-1}$]'}


pflux_labels = {'par_label':r'$F_\parallel / F^2$ ', 
                'perp_label':r'$F_\perp / F^2$',
                'phi_label':r'$F_\phi / F^2$'}

post_amp = "A10"
period = "p240"
tube_r = 'r30'
drivers = ['horiz', 'vert', 'Suni', 'Sarch', 'Slog']
exp_facs = [None, None, 'B0', 'B0005', 'B005']
captions = ['Horizontal', 'Vertical', 'Circular', 'Archemedian Spiral', 'Logarithmic Spiral']

figures = {}

for j, (driver, exp_fac, caption) in enumerate(zip(drivers, exp_facs, captions)):
    
    all_times, y, all_spoints = tdp.get_xy(ch4.data_dir, driver, period, post_amp, tube_r, exp_fac)
    data, beta_line = tdp.get_data(ch4.data_dir, driver, period, post_amp, tube_r, exp_fac)
    va_line, cs_line = tdp.get_speeds(ch4.data_dir, driver, period, post_amp, tube_r, exp_fac)

    Fdata, beta_line, avgs = tdp.get_flux(ch4.data_dir, driver, period, post_amp, tube_r, exp_fac)

    
    fd = lambda args: [a.T for a in args]
       
    ph.xxlim = -150
    
    fig, axes = plt.subplots(nrows=3, ncols=2, sharex=True, figsize=figsize)
    ph.triple_plot(axes[:,0], all_times, y, *fd(data), beta_line=1./beta_line,
                   levels=[1.,3.,5.,7.,10.,100.], manual_locations=False, **pvel_labels)
    
    
    
    ph.triple_plot(axes[:,1], all_times, y, *fd(Fdata), beta_line=1./beta_line,
                   levels=[1.,3.,5.,7.,10.,100.], manual_locations=False, cmap='PRGn', **pflux_labels)
    
    for ax in axes.flat:
        ph.add_phase_speeds(ax, all_times, y, va_line, cs_line, dx_scale=1e6, color='g')
        
    #Remove the top two x labels
    axes[0,0].set_xlabel('')
    axes[1,0].set_xlabel('')
    axes[0,1].set_xlabel('')
    axes[1,1].set_xlabel('')
    fig.tight_layout(h_pad=0.05)
    figures[driver] = fig
/opt/miniconda/envs/thesis/lib/python2.7/site-packages/matplotlib/__init__.py:1350: UserWarning:  This call to matplotlib.use() has no effect
because the backend has already been chosen;
matplotlib.use() must be called *before* pylab, matplotlib.pyplot,
or matplotlib.backends is imported for the first time.

  warnings.warn(_use_error_msg)

Vertical Driver

In [17]:
figures['vert']
Out[17]:

Horizontal Driver

In [18]:
figures['horiz']
Out[18]:

Logarithmic Spiral

In [19]:
figures['Slog']
Out[19]:

Comparison of Flux Averages

In [20]:
from flux_comparison import make_flux_bar_chart, get_averages

Favgs = get_averages(ch4.data_dir)

fig = make_flux_bar_chart((13,11), ch4.data_dir)
fig
Out[20]:

Effects of Logarithmic Spiral Expansion Factor

Chapter 5

$$ F(x) = A \frac{\cos(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{\sin(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ $$

where, $\theta = \tan^{-1}\left(\frac{y}{x}\right),\ \phi = \tan^{-1}\left(\frac{1}{B_L}\right)$

In [21]:
from sacconfig import SACConfig
cfg = SACConfig(cfg_file="./python/ch5_config.cfg")

BL = np.array([0.015, 0.05, 0.15, 0.45, 1.5])
In [22]:
fig, ax = plt.subplots(figsize=(14,2), dpi=600)
ax.plot(BL, np.ones(BL.size), 'x', markersize=10, mew=2)
ax.errorbar([0.15], [1], xerr=np.array([[-1*(0.15-1/(6.4-1.6)), 0.15+1/(6.4+1.6)]]).T, mew=2, elinewidth=2)
ax.semilogx()
ax.get_yaxis().set_visible(False)
ax.set_frame_on(False)
ax.get_xaxis().tick_bottom()
ax.xaxis.set_tick_params(width=2)
ax.xaxis.set_tick_params(width=2, which='minor')
ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
ax.xaxis.set_ticks(BL)
xmin, xmax, ymin, ymax = ax.axis()
ax.add_artist(plt.Line2D((xmin, xmax), (ymin, ymin), color='black', linewidth=1.4))
l = ax.set_xlim([0.01, 2.0])
l = ax.set_xlabel(r'$B_L$', fontsize=18)

fig.tight_layout(h_pad=0.01)
fig
Out[22]:
In [23]:
beta = False
def add_triple_phase(ax, tube_r):
    ps = ph.get_phase_speeds(cfg, tube_r)
    for ax0 in ax:
        ph.add_phase_speeds(ax0, color='g', **ps)

bl_figures = {}
for j, bl in enumerate(BL):
    cfg.exp_fac = bl
    
    fig, ax = plt.subplots(nrows=3, ncols=2, sharex=True, figsize=figsize)
    
    kwargs = ph.get_single_velocity(cfg, 'r30', beta=beta)
    kwargs.update(pvel_labels)
    
    ph.triple_plot(ax[:,0], **kwargs)
    
    kwargs = ph.get_single_percentage_flux(cfg, 'r30', beta=beta)
    kwargs.update(pflux_labels)
    kwargs.update({'cmap': 'PRGn'})
    
    ph.triple_plot(ax[:,1], **kwargs)
    
    #Remove the top two x labels
    ax[0,0].set_xlabel('')
    ax[1,0].set_xlabel('')
    add_triple_phase(ax[:,0], 'r30')
    ax[0,1].set_xlabel('')
    ax[1,1].set_xlabel('')
    add_triple_phase(ax[:,1], 'r30')
    #add_all_bpert(ax, 'r30')
    fig.tight_layout(h_pad=0.05)
    bl_figures['B{}'.format(str(bl).replace('.',''))] = fig

More Time-Distance Diagrams

$B=0.015$

In [24]:
bl_figures['B0015']
Out[24]:

$B=0.15$

In [25]:
bl_figures['B015']
Out[25]:

$B=1.5$

In [26]:
bl_figures['B15']
Out[26]:

Average Wave Flux

In [27]:
import os
int_periods = np.floor(600./cfg.period)*180

def calc_avgs(tube_r):
    AvgsP = np.zeros([3,len(BL)])
    for i, bl in enumerate(BL):
        cfg.exp_fac = bl
        
        times = np.load(os.path.join(cfg.data_dir, 'Times_{}.npy'.format(cfg.get_identifier())))
        max_index = np.argmin(np.abs(int_periods - times))
        
        Fpar, Fperp, Fphi = map(np.load, ph.glob_files(cfg, tube_r, 'LineFlux*Fp*npy'))
        #Fpar, Fperp, Fphi = map(np.load, ph.glob_files(cfg, tube_r, '*vp*npy'))
        Fpar[np.abs(Fpar)<1e-5], Fperp[np.abs(Fperp)<1e-5], Fphi[np.abs(Fphi)<1e-5] = 0., 0., 0.
        Fpar, Fperp, Fphi = Fpar[:max_index,:], Fperp[:max_index,:], Fphi[:max_index,:]
        
        Ftot2 = (Fpar**2 + Fperp**2 + Fphi**2)
        Fpar2, Fperp2, Fphi2 = np.array([Fpar, Fperp, Fphi])**2
        FparP, FperpP, FphiP = (Fpar2/Ftot2)*100, (Fperp2/Ftot2)*100, (Fphi2/Ftot2)*100
        
        FparP = np.mean(np.ma.masked_array(FparP, np.isnan(FparP)))
        FperpP = np.mean(np.ma.masked_array(FperpP, np.isnan(FperpP)))
        FphiP = np.mean(np.ma.masked_array(FphiP, np.isnan(FphiP)))
        
        AvgsP[:, i] = FparP, FperpP, FphiP
    return AvgsP

figsize = (9.5,11)
fig, axs = plt.subplots(nrows=3, figsize=figsize, sharex=True)
titles = ["Flux Surface Radius $=156$ km", "Flux Surface Radius $=468$ km", "Flux Surface Radius $=936$ km"]
tubes = []
for i, ax in enumerate(axs):
    AvgsP = calc_avgs(cfg.tube_radii[i])
    tubes.append(AvgsP)
    ax.semilogx()
    ax.plot(BL, AvgsP[0], 'o', label=r"$F_\parallel^2$", mew=0, ms=10)
    ax.plot(BL, AvgsP[1], '_', label=r"$F_\perp^2$", mew=2, ms=10)
    ax.plot(BL, AvgsP[2], 'x', label=r"$F_\phi^2$", mew=2, ms=10)
    ax.set_ylabel("% Square Wave Flux")
    ax.set_title(titles[i])
    ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
    ax.xaxis.set_ticks(BL)
    ax.set_xlim([0.01, 2.01])
    ax.set_ylim([0, 85])
    err = np.array([-1*(0.15-1/(6.4-1.6)), 0.15+1/(6.4+1.6)])
    ax.fill_betweenx(np.linspace(-5,105), err[0], err[1], alpha=0.3, color='green', linewidth=0)

axs[0].legend(loc=9)
axs[-1].set_xlabel("Logarithmic Spiral Expansion Factor ($B_L$)")

plt.tight_layout()
fig
Out[27]:

Effects of Period on MHD Wave Generation from a Logarithmic Spiral Driver

Chapter 6

Maintaining Consistent Energy Input


The amplitude is calculated using: $$A^2 \propto \frac{1}{P}$$ with $A=10$ $m/s$ and $P=240$ s as the reference point.
Period [seconds] Amplitude [ms$^{-1}$]
$30.0$ $20\sqrt{2}$
$60.0$ $20$
$90.0$ $20\sqrt{\frac{2}{3}}$
$120.0$ $10\sqrt{2}$
$150.0$ $4\sqrt{10}$
$180.0$ $\frac{20}{\sqrt{3}}$
$210.0$ $20\sqrt{\frac{2}{7}}$
$240.0$ $10$
$270.0$ $\frac{20}{3}\sqrt{2}$
$300.0$ $4\sqrt{5}$
In [28]:
from period_amps import periods, str_amps, sim_params
all_periods = sim_params[:10]
periods = periods[:10]

cfg = SACConfig(cfg_file='./python/ch6_config.cfg')
cfg.data_dir = '/home/stuart/GitHub/Thesis/thesis/Chapter6/Data/'
In [29]:
beta = False
cfg.exp_fac = 0.15
ph.xxlim = 600
tube_r = 'r30'

def add_all_bpert(ax, tube_r, N=4, levels=None):
    kwargs = ph.get_triple(cfg, beta=beta, single='bpert')
    x = kwargs['x_{}'.format(tube_r)]
    y = kwargs['y_{}'.format(tube_r)]
    par = kwargs['par_line_{}'.format(tube_r)].T[::-1, :]
    par[np.abs(par)<=1e-12] = 0
    perp = kwargs['perp_line_{}'.format(tube_r)].T[::-1, :]
    perp[np.abs(perp)<=1e-12] = 0
    phi = kwargs['phi_line_{}'.format(tube_r)].T[::-1, :]
    phi[np.abs(phi)<=1e-12] = 0
    ax[0].contour(x, y, par, N, colors='k', linewidths=np.linspace(0.5,1.5,N))
    ax[1].contour(x, y, perp, N, colors='k', linewidths=np.linspace(0.5,1.5,N))
    ax[2].contour(x, y, phi, N, colors='k', linewidths=np.linspace(0.5,1.5,N))	                   

def add_triple_phase(ax, tube_r):
    ps = ph.get_phase_speeds(cfg, tube_r)
    for ax0 in ax:
        ph.add_phase_speeds(ax0, color='g', **ps)

captions = {p: r"Period: ${}$ s amplitude: ".format(p) + a + r" ms$^{{-1}}$" for p, a in zip(periods, str_amps)[:10]}
#print(captions, file=sys.stderr)
width = 0.79

p_figures= {}
for i, paf in enumerate(all_periods):
    [setattr(cfg, f, getattr(paf, f)) for f in paf._fields]

    fig, ax = plt.subplots(nrows=3, ncols=2, figsize=(17,8), sharex=True)
    
    kwargs = ph.get_single_velocity(cfg, 'r30', beta=beta)
    kwargs.update(pvel_labels)
    
    ph.triple_plot(ax[:,0], **kwargs)
    
    kwargs = ph.get_single_percentage_flux(cfg, 'r30', beta=beta)
    kwargs.update(pflux_labels)
    kwargs.update({'cmap': 'PRGn'})
    
    ph.triple_plot(ax[:,1], **kwargs)
    
    #Remove the top two x labels
    ax[0,0].set_xlabel('')
    ax[1,0].set_xlabel('')
    add_triple_phase(ax[:,0], 'r30')
    ax[0,1].set_xlabel('')
    ax[1,1].set_xlabel('')
    add_triple_phase(ax[:,1], 'r30')
    #add_all_bpert(ax, 'r30')
    fig.tight_layout(h_pad=0.05)
    p_figures[str(paf.period)] = fig
/opt/miniconda/envs/thesis/lib/python2.7/site-packages/matplotlib/pyplot.py:516: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`).
  max_open_warning, RuntimeWarning)

Even More Time-Distance Diagrams

$P=30$ $s$ $A=20\sqrt{2}$ $m/s$

In [30]:
p_figures['30.0']
Out[30]:

$P=150$ $s$ $A=4\sqrt{10}$ $m/s$

In [31]:
p_figures['150.0']
Out[31]:

$P=300$ $s$ $A=4\sqrt{5}$ $m/s$

In [32]:
p_figures['300.0']
Out[32]:

Period Comparison

In [33]:
from period_amps import periods, sim_params
sim_params = sim_params[:10]
periods = np.array(periods[:10])

fig, axs = plt.subplots(nrows=3, figsize=(10,11), sharex=True)
titles = ["Flux Surface Radius $=156$ km", "Flux Surface Radius $=468$ km", "Flux Surface Radius $=936$ km"]
tubes = []
for i, ax in enumerate(axs):
    AvgsP = ph.get_all_avgs(cfg, cfg.tube_radii[i], sim_params)
    tubes.append(AvgsP)
    ax.plot(periods, AvgsP[0], 'o', label=r"$F_\parallel^2$", mew=0, ms=12)
    ax.plot(periods, AvgsP[1], '_', label=r"$F_\perp^2$", mew=2, ms=12)
    ax.plot(periods, AvgsP[2], 'x', label=r"$F_\phi^2$", mew=2, ms=12)
    ax.set_ylabel("% Square Wave Flux")
    ax.set_title(titles[i])
    ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
    ax.xaxis.set_ticks(periods)
    ax.set_xticklabels(["{}\n[{:n}]".format(p, 600//p) for p in periods])
    ax.set_ylim([10, 75])
    ax.set_xlim([25, 305])

axs[-1].set_xlabel("Period [s] \n [Number of periods averaged]")
axs[0].legend(loc=2)

#axs[0].legend(bbox_to_anchor=(1.06, 1.05))
plt.tight_layout(h_pad=0.1)
fig
Out[33]:

Conclusions

Future Work

Where we are going we don't need roads.

Bibliography